Linear Convergence of ADMM Under Metric Subregularity for Distributed Optimization

• 成果类型: Article
• 署名作者: Pan, Xiaowei; Liu, Zhongxin; Chen, Zengqiang
• 署名单位: Nankai University
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2022.3185178
• 发表日期: 2023
• 页码: 2513-2520
• 关键词: convergence; Convex functions; measurement; optimization; cost function; Symmetric matrices; simulation; Alternating direction method of multipliers (ADMM); Composite optimization; distributed optimization; linear rate; metric subregularity; nonergodic

摘要

The alternating direction method of multipliers (ADMM) has seen much progress in the literature in recent years. Usually, linear convergence of distributed ADMM is proved under either second-order conditions or strong convexity. When both conditions fail, an alternative is expected to play the role. In this article, it is shown that distributed ADMM can achieve a linear convergence rate by imposing metric subregularity on a defined mapping. Furthermore, it is proved that both second-order conditions and strong convexity imply metric subregularity under reasonable conditions, e.g., the cost functions being twice continuously differentiable in a neighborhood. In addition, nonergodic convergence rates are presented as well for problems under consideration. Finally, simulation results are carried out to illustrate the efficiency of the proposed algorithm.